A survey organization takes a simple random sample of 625 households from a city of 80,000 households. On the average, there are 2.30 persons per sample household, and the SD is 1.75. Say whether each of the following statements are True or false and explain
(a) The SE for the sample average is 0.07
(b) A 95% confidence interval for the average household size in the sample is 2.16 to 2.44
(c)A 95% confidence interval for the average household size in the city is 2.16 to 2.44
(d)A 95% confidence interval of the household size in the city contain btwn 2.16 to 2.44
persons
SEm = SD/√n
true
(a) The SE for the sample average is 0.07.
True. The standard error (SE) measures the variability or precision of the sample mean estimate. In this case, the formula for the SE is:
SE = (standard deviation of the sample) / sqrt(sample size)
Given that the standard deviation is 1.75 and the sample size is 625, we can calculate the SE as:
SE = 1.75 / sqrt(625) = 1.75 / 25 = 0.07. Therefore, the statement is true.
(b) A 95% confidence interval for the average household size in the sample is 2.16 to 2.44.
True. The confidence interval is a range of values within which the true population parameter is likely to fall. For a 95% confidence interval, we use the formula:
Confidence interval = (sample mean) ± (margin of error)
The margin of error is calculated by multiplying the standard error by the critical value, which depends on the confidence level and the sampling distribution (such as the t-distribution for small samples).
Given that the sample mean is 2.30 and the standard error is 0.07, we can calculate the margin of error using the appropriate critical value. Assuming the critical value corresponds to a 95% confidence level, the interval would indeed be 2.16 to 2.44. Therefore, the statement is true.
(c) A 95% confidence interval for the average household size in the city is 2.16 to 2.44.
False. The confidence interval calculated in part (b) represents an estimate for the average household size in the sample, not in the entire city. Without information on the population parameters, we cannot directly calculate the confidence interval for the city itself. Therefore, this statement is false.
(d) A 95% confidence interval for the household size in the city contains between 2.16 to 2.44 persons.
We cannot determine the correctness of this statement without knowing the population parameters or conducting further analysis. The confidence interval estimated in part (b) is specific to the sample and does not directly provide information about the population parameter or the entire city.
a) True
b) False
c) True
d) false